Question

The value of \(cos15^{\circ}-sin15^{\circ}\) is ......

• A. \(\sqrt{2}\)
• B. \(-\sqrt{2}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. \(-\frac{1}{\sqrt{2}}\)

Hint:

Write cos x minus sin x as root 2 cos(x + 45 degrees).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
An expression \(\cos x - \sin x\) can be written as a single cosine using the amplitude-phase form.

Step 2: Key Identity:
\[ \cos x - \sin x = \sqrt{2}\cos\left(x + 45^{\circ}\right) \]

Step 3: Detailed Explanation:
Take \(x = 15^{\circ}\):
\[ \cos 15^{\circ} - \sin 15^{\circ} = \sqrt{2}\cos 60^{\circ} = \sqrt{2} \times \frac{1}{2} = \frac{1}{\sqrt{2}} \]
Check numerically: \(\cos 15^{\circ} = 0.9659\) and \(\sin 15^{\circ} = 0.2588\), so the difference is \(0.7071 = 1/\sqrt2\).

Step 4: Why the other options are wrong.
\(\sqrt2 = 1.414\) is too large because both values are below 1. The negative options are impossible since \(\cos 15^{\circ} > \sin 15^{\circ}\), so the difference is positive.

Final Answer:
The value is \(\frac{1}{\sqrt{2}}\), option (C). \[ \boxed{\frac{1}{\sqrt{2}}} \]